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Math-2561-A1 · revision r1 · on Math-2561

OpenAI claim summary for Math-2561

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 035, 2026-10-05): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Numerical-dimension-one log abundance in characteristic above three
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Numerical-dimension-one log abundance in characteristic above three

OpenAI's claim

For projective log-canonical threefold pairs in characteristic $p>3$, a nef rational adjoint of numerical dimension one is claimed to be semiample without terminality or factoriality assumptions. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Over an algebraically closed field of characteristic $p>3$, take a projective log canonical threefold pair $(X,B)$ with normal $X$, effective rational $B$ and rational Cartier nef $D=K_X+B$. If for an ample Cartier $H$, \[ DH^2>0,\qquad D^2H=0, \] so $D$ has numerical dimension one, OpenAI claims $D$ semiample. This requires the stated prime range and numerical-dimension restriction.

Outline of the argument

Manuscript 1 (family 035, 2026-10-05): Boundary intersection control separates components from a primitive nef cycle by quantitative bend-and-break. The resulting connected Cartier divisor meets the companion cycle-obstruction hypotheses, ruling out the remaining zero-Kodaira case and giving semiampleness.

Formal verification

Family 035 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources